How to graph alternative solution of a function

HI there, pretty simple question and more of a maths one than a technical MATLAB one.

For each value of Launch Angle there are two solutions for the value of A. I'm plotting one at the moment and want to plot the other. e.g. at launch angle of 10deg, A can equal both 0.21 and 2.2ish

How do I change my code to plot the other solution?

A_values = ones(1,45);
B_values = ones(1,45);
LaunchAngle = ones(1, 45);
for qL = 1:45
   LaunchAngle(1,qL) = qL;
   qLrad = deg2rad(qL);
   Rd = 1.7;   % Y asymptote of curve
   XL = 0;  % Release point x
   X0 = 1.7; % Starting point x
   F = @(A,B) [((-Rd + ((2*Rd) /(1 + exp(A*(X0 - B)))))), (((-2*A*Rd*exp(A*(XL-B)))/((exp(A*(XL-B))+1)^2))-tan(qLrad))];
   AB0 = [1; 1]*1j;
   AB = fsolve(@(b)F(b(1),b(2)), AB0);
   A_values(:,qL) = AB(1)*-1;
   B_values(:,qL) = AB(2);
end

plot(LaunchAngle, A_values, LaunchAngle, B_values);
xlabel('Launch angle (deg)');
ylabel('Required trajectory function parameters');
title('Required trajectory function parameters for launch point x = 0');
legend('A', 'B');

Answer 1

[Deepak]

Hi @Adam Thompson,

We can plot both possible solutions of (A) against the launch angles by using two distinct initial guesses for the “fsolve” function. By setting different initial conditions, such as AB0_1 and AB0_2, we guide “fsolve” to explore separate regions of the solution space, thereby identifying both solutions.

Next, we can store these solutions in separate arrays and plots them against the launch angles, allowing for a clear visualization of how each solution for (A) varies with the launch angle. This method ensures that both trajectories are represented, providing a comprehensive understanding of the behaviour of model.

Here is the MATLAB code to accomplish the same:

A_values1 = ones(1,45); 
B_values1 = ones(1,45); 
A_values2 = ones(1,45); 
B_values2 = ones(1,45); 
LaunchAngle = ones(1, 45); 

for qL = 1:45 
    LaunchAngle(1,qL) = qL; 
    qLrad = deg2rad(qL); 
    Rd = 1.7;   % Y asymptote of curve 
    XL = 0;  % Release point x 
    X0 = 1.7; % Starting point x 
    F = @(A,B) [((-Rd + ((2*Rd) /(1 + exp(A*(X0 - B)))))), (((-2*A*Rd*exp(A*(XL-B)))/((exp(A*(XL-B))+1)^2))-tan(qLrad))]; 
    
    % First solution 
    AB0_1 = [1; 1]*1j; 
    AB1 = fsolve(@(b)F(b(1),b(2)), AB0_1); 
    A_values1(:,qL) = AB1(1)*-1; 
    B_values1(:,qL) = AB1(2); 
    
    % Second solution 
    AB0_2 = [-2; -2]*1j;   
    AB2 = fsolve(@(b)F(b(1),b(2)), AB0_2); 
    A_values2(:,qL) = AB2(1)*-1; 
    B_values2(:,qL) = AB2(2); 
end 

figure; 
plot(LaunchAngle, A_values1, 'b', LaunchAngle, A_values2, 'r'); 
xlabel('Launch angle (deg)'); 
ylabel('A values'); 
title('Two solutions for A at each launch angle'); 
legend('Solution 1', 'Solution 2'); 

figure; 
plot(LaunchAngle, B_values1, 'b', LaunchAngle, B_values2, 'r'); 
xlabel('Launch angle (deg)'); 
ylabel('B values'); 
title('Two solutions for B at each launch angle'); 
legend('Solution 1', 'Solution 2'); 

Please find attached the documentation of functions used for reference:

fsolve: www.mathworks.com/help/optim/ug/fsolve.html

plot: www.mathworks.com/help/matlab/ref/plot.html

I hope this helps in resolving the issue.